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Currently, Google shows www.widgets.com/blah in SERPS.
Since my root domain of www.widgets.com will not be changing, will my PR and position in SERPS be affected once Google starts indexing my new site at www.widgets.com (using an Apache redirect from old site to new)? Is it like changing to a totally new domain even though I'm just dropping the /blah? My internal page names are changing too, but my main concern is my homepage.
A 301 redirect is probably your best approach (site search [searchengineworld.com] should offer much discussion on the topic). Hopefully, you'll get the backlinks credited to the new address.